Monk rules for type $GL_n$ Macdonald polynomials

TL;DR

This paper derives Monk rules for Macdonald polynomials using Cherednik's intertwiners, connecting previous formulas for Schubert and Iwahori-spherical polynomials, and extends to special cases like key polynomials.

Contribution

Introduces a new derivation of Monk rules for Macdonald polynomials via Cherednik's intertwiners, linking prior work and extending to special polynomial cases.

Findings

Monk rules for Macdonald polynomials derived using Cherednik's intertwiners.

Specializations yield Monk rules for Iwahori-spherical and key polynomials.

Connects Baratta's formulas with Yip's product formulas.

Abstract

In this paper we give Monk rules for Macdonald polynomials which are analogous to the Monk rules for Schubert polynomials. These formulas are similar to the formulas given by Baratta (2008), but our method of derivation is to use Cherednik's interwiners. Deriving Monk rules by this technique addresses the relationship between the work of Baratta and the product formulas of Yip (2010). Specializations of the Monk formula's at $q=0$ and/or $t=0$ provide Monk rules for Iwahori-spherical polynomials and for finite and affine key polynomials.